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Win money with magic squares

Submitted by Marianne on May 4, 2010

Leonhard Euler, 1707-1783.

Magic squares have been known and studied for many centuries, but there are still surprisingly many unanswered questions about them. In an effort to make progress on these unsolved problems, twelve prizes totalling €8,000 and twelve bottles of
champagne have now been offered for the solutions to twelve magic square enigmas.

A magic square consists of whole numbers arranged in a square, so that all rows, all columns and the two diagonals sum to the same number. An example is the following 4×4 magic square, consisting entirely of square numbers, which the mathematician Leonhard Euler sent to Joseph-Louis Lagrange in
1770:

682

292

412

372

172

312

792

322

592

282

232

612

112

772

82

492

A 4×4 magic square of squares by Euler. An n×n magic square uses n2 distinct integers and has the same sum S for its n rows, its n columns and its two diagonals. Here S=8,515.

It's still not known whether a 3×3 magic squares consisting entirely of squares is possible.

The prize money and champagne will be divided between the people who send in first solutions to one of the six main enigmas or the six smaller enigmas listed below. Solutions should be sent to Christian Boyer. His website gives more information about every enigma, and will
contain regular updates regarding received progress and prizes won.

Note that the enigmas can be mathematically rewritten as sets of Diophantine equations: for example, a 3×3 magic
square is a set of eight equations (corresponding to the three rows, three columns and two diagonals) in ten unknowns (the nine entries and the magic constant to which each line sums).

Here are the six main and six small enigmas:

How big are the smallest possible magic squares of squares: 3×3 or 4×4?

In 1770 Leonhard Euler was the first to construct 4×4 magic squares of squares, as mentioned above. But
nobody has yet succeeded in building a 3×3 magic square of squares or proving that it is impossible.
Edouard Lucas worked on the subject in 1876. Then, in 1996, Martin Gardner offered $100 to the first person
who could build one. Since this problem — despite its very simple appearance — is incredibly difficult to solve
with nine distinct squared integers, here is a question which should be easier:

Main Enigma 1 (€1000 and 1 bottle):
Construct a 3×3 magic square using seven (or eight, or nine)
distinct squared integers different from the only known example and its rotations, symmetries and k2 multiples. Or prove that it is impossible.

3732

2892

5652

360721

4252

232

2052

5272

222121

The only known example of a 3×3 magic square using seven distinct squared integers, by Andrew Bremner. S=541,875.

How big are the smallest possible bimagic squares: 5×5 or 6×6?

A bimagic square is a magic square which stays magic after squaring its integers. The first known were
constructed by the Frenchman G. Pfeffermann in 1890 (8×8) and 1891 (9×9). It has been proved that 3×3 and 4×4 bimagics
are impossible. The smallest bimagics currently known are 6×6, the first one of
which was built in 2006 by Jaroslaw Wroblewski, a mathematician at Wroclaw University, Poland.

17

36

55

124

62

114

58

40

129

50

111

20

108

135

34

44

38

49

87

98

92

102

1

28

116

25

86

7

96

78

22

74

12

81

100

119

A 6×6 bimagic square by Jaroslaw Wroblewski. S1=408, S2=36,826.

Main Enigma 2 (€1000 and 1 bottle): construct a 5×5 bimagic square using distinct positive
integers, or prove that it is impossible.

How big are the smallest possible semi-magic squares of cubes: 3×3 or 4×4?

An n×n semi-magic square is a square whose n rows and n columns have the same sum, but whose
diagonals can have any sum. The smallest semi-magic squares of cubes currently known are 4×4
constructed in 2006 by Lee Morgenstern, an American mathematician. We also know 5×5 and 6×6 squares,
then 8×8 and 9×9, but not yet 7×7.

163

203

183

1923

1803

813

903

153

1083

1353

1503

93

23

1603

1443

243

A 4×4 semi-magic square of cubes by Lee Morgenstern. S=7,095,816.

Main Enigma 3 (€1000 and 1 bottle): Construct a 3×3 semi-magic square using positive distinct
cubed integers, or prove that it is impossible.

Small Enigma 3a (€100 and 1 bottle): Construct a 7×7 semi-magic square using positive distinct
cubed integers, or prove that it is impossible.

How big are the smallest possible magic squares of cubes: 4×4, 5×5, 6×6, 7×7 or 8×8?

The first known magic square of cubes was constructed by the Frenchman Gaston Tarry in 1905, thanks to a
large 128×128 trimagic square (magic up to the third power). The smallest currently known magic squares of
cubes are 8×8 squares constructed in 2008 by Walter Trump, a German teacher of mathematics. We do not
know any 4×4, 5×5, 6×6 or 7×7 squares. It has been proved that 3×3 magic squares of cubes are impossible.

113

93

153

613

183

403

273

683

213

343

643

573

323

243

453

143

383

33

583

83

663

23

463

103

633

313

413

303

133

423

393

503

373

513

123

63

543

653

233

193

473

363

433

333

293

593

523

43

553

533

203

493

253

163

53

563

13

623

263

353

483

73

603

223

An 8×8 magic square of cubes by Walter Trump. S=636,363.

Main Enigma 4 (€1000 and 1 bottle): Construct a 4×4 magic square using distinct positive cubed
integers, or prove that it is impossible.

Small Enigma 4a (€500 and 1 bottle): Construct a 5×5 magic square using distinct positive cubed
integers, or prove that it is impossible.

Small Enigma 4b (€500 and 1 bottle): Construct a 6×6 magic square using distinct positive cubed
integers, or prove that it is impossible.

Small Enigma 4c (€200 and 1 bottle): Construct a 7×7 magic square using distinct positive cubed
integers, or prove that it is impossible. (When such a square is constructed, if small enigma 3a of
the 7×7 semi-magic is not yet solved, then the person will win both prizes — that is to say a total of
€300 and 2 bottles.)

How big are the smallest integers allowing the construction of a multiplicative magic cube?

Contrary to all other enigmas which concern the magic squares, this one concerns magic cubes. An
n×n×n multiplicative magic cube is a cube whose n2 rows, n2 columns, n2 pillars, and 4 main diagonals
have the same product P. Today the best multiplicative magic cubes known are 4×4×4 cubes in which the largest
used number among their 64 integers is equal to 364. We do not know if it is possible to construct a cube
with smaller numbers.

Main Enigma 5 (€1000 and 1 bottle): Construct a multiplicative magic cube in which the distinct
positive integers are all strictly lower than 364. The size is free: 3×3×3, 4×4×4, 5×5×5,... . Or prove
that it is impossible.

An n×n additive-multiplicative magic square is a square in which the n rows, n columns and two diagonals have
the same sum S, and also the same product P. The smallest known are 8×8 squares, the first one of which
was constructed in 1955 by Walter Horner, an American teacher of mathematics. We do not know any 5×5,
6×6 or 7×7 squares. It has been proved that 3×3 and 4×4 additive-multiplicative magic squares are impossible.

I've pieced together a small script in Excel searching for a perfect 3x3 magic square of squares.
The first version was limited to search for integers between 1 and 1000. But I can't be the first one trying to brute force the main enigma 1 problem here. Not when there is a bottle at stake. But how far up in the integers have we searched for a solution?

It has been mathematically proven that the smallest number in a complete 3x3 magic square of squares – if it even exists – would have to be larger than 10^14.

I would definitely recommend starting with the algebra before moving on to brute force. Personally, I've found a parametric formula which turns 4 variables of almost any arbitrary value into a 5/9 square of squares (the 4 corners and the center), and I've brute forced about 150 specific values which create 6/9 squares (the 4 corners, the middle, and one of the sides).

Out of all of the 6/9 squares that I have found myself, my favorite (per my love of horrifyingly large numbers) is the one with

Yes.
Because: Main Enigma 1 (€1000 and 1 bottle): Construct a 3×3 magic square using seven (or eight, or nine) distinct squared integers different from the only known example and its rotations, symmetries and k2 multiples. Or prove that it is impossible.

I checked all solutions up tot S=1,500,000. Did not find another 3 x 3 magic square of 7 or more distinct squared integers. I did however find many with 6. They seem to get more rare the higher you get. The last one I found was: